Math 145, Winter 99: Chaos Theory
Lab #2, Week 03
Population dynamics, an exemplary 1D flow

We introduce 2D flows in the Stella environment, and in the Maple environment, with three examples:
  • 2D population dynamics, Volterra-Lotka #1:
    rabbits and sheep
  • 2D population dynamics, Volterra-Lotka #2:
    big fish and small fry
  • Van der Pol oscillator with Hopf bifurcation

2D Population dynamics

1. Rabbits and Sheep
Another Volterra-Lotka system, as described in Strogatz, Sec. 6.4, p. 155. The equations are:
  • x' = x(3-x-2y)
  • y' = y(2-x-y)
where x is the population of rabbits, and y that of sheep
Download: [Stella model]
Hint: Here is the skeleton of the portrait, see Strogatz, p. 159.

Recall our color code:
  • red points are attractors
  • black points are saddles
  • green points are repellors
  • insets are green (grin)
  • outsets are blue (blout)
2. Big fish and small fry
The classical Volterra-Lotka system of 1924, as described in Abraham and Shaw (DGB, 1st edn., Part I, Sec. 2.4 and p. 212). See also Strogatz, Ch. 6, p. 189, and Hirsch and Smale.
The equations are:
  • x' = x(1-y)
  • y' = ry(x-1)
where x is the population of small fry, and y that of big fish.
Download: [Maple work sheet]
Download: [Maple script (text)]

Van der Pol scheme

The equations are:
  • x' = y
  • y' = -x + y(1-x2)
where x is the current and y the voltage in a triode oscillator. See Strogatz, pp. 181, 198, 211, 228, for the equations. See Hirsch and Smale for a nice proof.
Download: [Stella model]

Recommended experiments

  • Find all attractors, basins, and separatrices
  • Vary parameters and identify bifurcations

References

  • Abraham and Shaw
  • Strogatz
  • Hirsch and Smale
Revised 12 Feb 1999 by Ralph